Optimal. Leaf size=47 \[ -\frac {2 a^3}{3 x^{3/2}}-\frac {6 a^2 b}{\sqrt {x}}+6 a b^2 \sqrt {x}+\frac {2}{3} b^3 x^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {43} \[ -\frac {6 a^2 b}{\sqrt {x}}-\frac {2 a^3}{3 x^{3/2}}+6 a b^2 \sqrt {x}+\frac {2}{3} b^3 x^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{x^{5/2}} \, dx &=\int \left (\frac {a^3}{x^{5/2}}+\frac {3 a^2 b}{x^{3/2}}+\frac {3 a b^2}{\sqrt {x}}+b^3 \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^3}{3 x^{3/2}}-\frac {6 a^2 b}{\sqrt {x}}+6 a b^2 \sqrt {x}+\frac {2}{3} b^3 x^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.81 \[ \frac {2 \left (-a^3-9 a^2 b x+9 a b^2 x^2+b^3 x^3\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 34, normalized size = 0.72 \[ \frac {2 \, {\left (b^{3} x^{3} + 9 \, a b^{2} x^{2} - 9 \, a^{2} b x - a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 34, normalized size = 0.72 \[ \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 6 \, a b^{2} \sqrt {x} - \frac {2 \, {\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.72 \[ -\frac {2 \left (-b^{3} x^{3}-9 a \,b^{2} x^{2}+9 a^{2} b x +a^{3}\right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 34, normalized size = 0.72 \[ \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 6 \, a b^{2} \sqrt {x} - \frac {2 \, {\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 35, normalized size = 0.74 \[ -\frac {2\,a^3+18\,a^2\,b\,x-18\,a\,b^2\,x^2-2\,b^3\,x^3}{3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 46, normalized size = 0.98 \[ - \frac {2 a^{3}}{3 x^{\frac {3}{2}}} - \frac {6 a^{2} b}{\sqrt {x}} + 6 a b^{2} \sqrt {x} + \frac {2 b^{3} x^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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